Interest Rates and Forward Rate Agreements

Read Hull Chapter 4


  • Interest rate types
  • Measuring interest rates
  • Continuous compounding
  • Zero rates / zero curve
  • Bootstrap
  • Forward rate
  • Forward rate agreement (FRA)
  • Term structure of interest rates

Interest Rate Types

  • Common interest rates
    • Treasury rates
      • Rates on government instruments in its own currency
    • London Interbank Offered Rate (LIBOR) rates
      • LIBOR is the rate of interest at which a bank is prepared to deposit money with another bank.
      • The second bank typically possesses an AA rating
      • The British Bankers Association compiles LIBOR once a day on all major currencies for maturities up to 12 months
    • The London Interbank Bid Rate (LIBID)
      • LIBID is the rate which a AA bank is prepared to pay on deposits from another bank
    • Repurchase agreement (Repo) rates
      • Repurchase agreement is an agreement where a financial institution that owns securities agrees to sell them today for X and buy them back in the future for a slightly higher price, Y
      • Invented to circumvent regulations
      • U.S. banks could not pay interest on business checking accounts
      • The financial institution obtains a temporary loan and the securities become the collateral
      • The interest equals Y – X
      • Known as the repo rate
    • The Risk-Free Rate
      • Derivatives practitioners traditionally use the short-term risk-free rate, LIBOR
      • Many consider the Treasury rate is to be artificially low
        • Regulations may require banks to hold highly-liquid securities
        • Treasuries may have tax advantages
        • Government has power to tax if it experiences budget problems
      • As will be explained in a later lecture
        • Financial wizards use Eurodollar futures and swapsto extend the LIBOR yield curve beyond one year
        • The overnight indexed swap rate is increasingly being used instead of LIBOR as the risk-free rate
          • Credit markets for LIBOR froze during the 2008 Global Financial Crisis

Measuring Interest Rates

  • Compounding frequency
    • The compounding frequency for an interest rate is the unit of measurement
    • The difference between quarterly and annual compounding is analogous to the difference between miles and kilometers
  • Impact of Compounding
    • When we compound m times per year at rate R, an amount PV grows to FV=PV(1+R/m)m in one year
Compounding frequency Value of $100 in one year at 10%
Annual (m=1) 110.00
Semiannual (m=2) 110.25
Quarterly (m=4) 110.38
Monthly (m=12) 110.47
Weekly (m=52) 110.51
Daily (m=365) 110.516
Continuous (m→∞) 110.517


Continuous Compounding

  • In the limit, we obtain continuously compounded interest rates as we compound more and more frequently
    • $100 grows to $100eRT when invested at a continuously compounded rate R for time T
    • $100 received at time T discounts to $100e-RT at time zero when the continuously compounded discount rate is R
  • Discrete
    • Future value (FV)
    • Present value (PV)
      • FV=PV(1 + R/m)m t
      • m – how many payments per year
  • Continuous compounding
    • m→∞

continuous compounding

  • Operations with e
    • Future value
      • eRt similar to (1 + R)t
    • Present value
      • e-Rt = 1 / eRT similar to (1+R)-t = 1 / (1+R)t
    • Derivatives and Forward Rate Agreements (FRA) use continuous interest rates
  •  Conversion Formulas
    • Continuously compounded ratem, Rc
    • Same rate with compounding m times per year, Rm
      • Rc=m ln(Rm / m+1)
      • Rm=m (eRc / m–1)
    • Examples
      • 10% with semiannual compounding equals 2 ln( 0.1 / 2 +1 ) = 9.758% with continuous compounding
      • 8% with continuous compounding equals 4 ( e0.08/41 ) = 8.08% with quarterly compounding
      • Rates used in option pricing are nearly always expressed with continuous compounding
    • Easy to derive formula
      • Set FV = PV (1 + Rm/m)t m = PV et Rc
    • Trick question on exams
      • Problem gives interest rates that are not continuously compounded
      • Must convert interest rates to continuous compounding

Zero Rates / Zero Curve

  • A zero rate (or spot rate), for maturity T is the rate of interest earned on an investment that provides a payoff only at time T
    • Discount bonds
  • Zero curve – treats bonds as if they make one payment at maturity
    • Must convert coupon bonds into discount bonds or zero coupon bonds
  • Example of bond pricing is below
Zero Rate
(% cont. comp.)
0.5 5.0
1.0 5.8
1.5 6.4
2.0 6.8
  • Bond Pricing
    • To calculate the cash price of a bond we discount each cash flow at the appropriate zero rate
      • A bond has cash flows at different times
      • Use appropriate interest rate to calculate present value
      • In our example, the theoretical price of a two-year bond providing a 6% coupon semiannually is



  • Bond Yield
    • The bond yield is the discount rate that makes the present value of the cash flows on the bond equal to the market price of the bond
    • Suppose that the market price of the bond in our example equals its theoretical price of 98.39
    • The bond yield is solved by finding the value for y with continuous compounding
    • Can solve iteratively
      • Set y to any number such as y =0.1
      • If the PV of cashflows equal 98.39, then you are done.
      • If PV is higher than 98.39, then select a lower number such as y = 0.08
      • Then repeat until PV of cash flows equals 98.39
      • Excel goal seeker can solve these problems


  • Par Yield
    • The par yield for a certain maturity is the coupon rate that causes the bond price to equal its face value.
    • In our example we solve for c, which equals c=6.87, semi-annual compounding

par yield

  • Par Yield
    • If m is the number of coupon payments per year
    • d is the present value of $1 received at maturity
    • A is the present value of an annuity of $1 on each coupon date

par yield

  • Example
    • m = 2
    • d = 0.87284
    • A = 3.70027
  • Refer to derivation below. Notice where the terms m, d, and A come from.

par yield

The Bootstrap Method

  • Data to Determine Zero Curve
  • Half the stated coupon is paid each year*
Bond Principal Time to Maturity (yrs) Coupon per year ($)* Bond price ($)
100 0.25 0 97.5
100 0.50 0 94.9
100 1.00 0 90.0
100 1.50 8 96.0
100 2.00 12 101.6
  • Bootstrap – method to fill in missing values
  • The bond holder earns $ 2.5 on the 97.5 during 3 months.
    • The 3-month interest rate is 2.5/97.5 = 2.56
    • Convert to annual with quarterly compounding 2.56% x 4 = 10.256%
    • This is 10.127% with continuous compounding
  • Similarly the 6 month and 1 year rates are 10.469% and 10.536% with continuous compounding
  • To calculate the 1.5 year rate we solve

104e-1.5R)= / 104 = 88.604 / 104
ln( e-1.5R) = ln (0.8520)
-1.5R = -0.1602
R = 0.1068

  • R = 0.10681 or 10.681%
  • Similarly the two-year rate is 10.808%
  • Zero Curve Calculated from the Data

zero curve

Forward Rates

  • The forward rate is the future zero rate implied by today's term structure of interest rates
    • Term structure of interest rates is what we calculated in last slide
  • Formula
    • Suppose that the zero rates for time periods T1 and T2 are R1 and R2 with both rates continuously compounded.
    • Note – forward rate is a contract for an interest rate for a future time period between T1 and T2.
    • The formula becomes an approximation when rates are not expressed as continuous compounding
    • The forward rate for the period between times T1 and T2 is

forward rate

  • Calculate the forward rates for the interest rates in the table
Zero rate for n-year investment
(% per annum)
Forward rate for nth year
(% per annum)
1 3.0 -
2 4.0 5.0
3 4.6 5.8
4 5.0 6.2
5 5.5 7.5
  • For an upward sloping yield curve:
    • Forward Rate > Zero Rate > Par Yield
  • For a downward sloping yield curve
    • Par Yield > Zero Rate > Forward Rate

Forward Rate Agreement

  • A forward rate agreement (FRA) is an agreement that a certain interest rate will apply to a certain principal during a certain future time period
    • Example – you are buying or selling an interest rate for a specific time period
    • Since interest rates do not have a physical existence, parties settle differences in cash
  • An FRA is equivalent to an agreement where interest at a predetermined (fixed) rate, RK is exchanged for interest at the market rate
    • An FRA can be valued by assuming that the forward LIBOR interest rate, RF, will be realized
    • We are using the forward rate to make a prediction
    • How else could we place a value on a future interest we do not know
    • Thus, the value of an FRA is the present value of the difference between one party receiving a variable interest rate RF and paying the fixed rate RK
      • The counterparty does the opposite
      • The parties settle the FRA at the beginning when the FRA starts
      • Zero sum game
  • Example
    • A company has agreed to receive 4% on $100 million for 3 months starting in 3 years
      • Rate is fixed
    • The forward rate for the period between 3 and 3.25 years is 3%
      • Forward rate forecasts the variable rate
      • FVFRA=(0.04/4 – 0.03/4)x$100 million = $250,000
      • The value of the contract to the company is +$250,000 at 3.25 years
      • Discount today to get present value
    • Suppose rate proves to be 4.5% (with quarterly compounding
      • We know the real variabe rate
      • FVFRA= (0.04/4 – 0.045/4)x$100 million = -$125,000
      • The payoff is –$125,000 at the 3.25 year point
      • This is equivalent to a payoff of –$123,609 at the 3-year point
        • -$125,000 e-0.04475x0.25 = -$123,609.36
        • Continuous interest rate = 0.04475
        • Remember - parties settle FRA at the start of the FRA in cash
        • We are at year 3.25 and discount back by 0.25
        • Could also use -$125,000 / (1 + 0.045 / 4)0.25

Term Structure of Interest Rates

  • Theories try to explain why the yield curve is usually upward sloping
    • Expectations Theory:
      • Long-term interest rates should reflect expected short-term interest rates
      • forward rates equal expected future zero rates
    • Market Segmentation:
      • Short, medium and long rates are separate markets with their own independent supply and demand
    • Liquidity Preference Theory:
      • Investors prefer to invest with liquid securities but will invest in longer maturities with a premium
      • Forward rates higher than expected future zero rates
      • Suppose that the outlook for rates is flat and you have been offered the following choices
  • Which rate would you choose as a depositor? Which maturity for your mortgage?
Maturity Deposit rate Mortgage rate
1 year 3% 6%
5 year 3% 6%
  • Liquidity Preference Theory continued
    • To match the maturities of borrowers and lenders a bank has to increase long rates above expected future short rates
    • In our example the bank might offer
Maturity Deposit rate Mortgage rate
1 year 3% 6%
5 year 4% 7%